Generalized gamma distribution
Description
Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the parameterisation originating from Prentice (1974). Also known as the (generalized) log-gamma distribution.
Usage
1 2 3 4 5 6 |
Arguments
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
mu |
Vector of “location” parameters. |
sigma |
Vector of “scale” parameters. Note the inconsistent
meanings of the term “scale” - this parameter is analogous to the
(log-scale) standard deviation of the log-normal distribution,
“sdlog” in |
Q |
Vector of shape parameters. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P(X <= x), otherwise, P(X > x). |
Details
If g ~ Gamma(Q^{-2}, 1) , and w = log(Q^2*g) / Q, then x = exp(mu + sigma w) follows the generalized gamma distribution with probability density function
f(x | mu, sigma, Q) = |Q| (Q^{-2})^{Q^{-2}} / / (sigma * x * Gamma(Q^{-2})) exp(Q^{-2}*(Q*w - exp(Q*w)))
This parameterisation is preferred to the original parameterisation of
the generalized gamma by Stacy (1962) since it is more numerically
stable near to Q=0 (the log-normal distribution), and allows Q<=0.
The original is available in this package as
dgengamma.orig
, for the sake of completion and
compatibility with other software - this is implicitly restricted to
Q
>0 (or k
>0 in the original notation). The parameters of
dgengamma
and dgengamma.orig
are related as
follows.
dgengamma.orig(x, shape=shape, scale=scale, k=k) =
dgengamma(x, mu=log(scale) + log(k)/shape, sigma=1/(shape*sqrt(k)), Q=1/sqrt(k))
The generalized gamma distribution simplifies to the gamma, log-normal and Weibull distributions with the following parameterisations:
dgengamma(x, mu, sigma, Q=0) | = | dlnorm(x, mu, sigma) |
dgengamma(x, mu, sigma, Q=1) | = | dweibull(x, shape=1/sigma, scale=exp(mu)) |
dgengamma(x, mu, sigma, Q=sigma) | = | dgamma(x, shape=1/sigma^2, rate=exp(-mu) / sigma^2) |
The properties of the generalized gamma and its applications to survival analysis are discussed in detail by Cox (2007).
The generalized F distribution GenF
extends the
generalized gamma to four parameters.
Value
dgengamma
gives the density, pgengamma
gives the distribution
function, qgengamma
gives the quantile function, rgengamma
generates random deviates, Hgengamma
retuns the cumulative hazard
and hgengamma
the hazard.
Author(s)
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
References
Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539-544.
Farewell, V. T. and Prentice, R. L. (1977). A study of distributional shape in life testing. Technometrics 19(1):69-75.
Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409-419.
Cox, C., Chu, H., Schneider, M. F. and Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine 26:4252-4374
Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:1187-92
See Also
GenGamma.orig
, GenF
,
Lognormal
, GammaDist
, Weibull
.